in an iterative fashion. Here is a step length. Now use the
updated
to calculate the new residuals
and new raypath
matrix ,
and use equation 2.9 to update the slowness perturbation. Repeat this
procedure until convergence. This is the method of steepest descent,
and usually books do not recommend its use due to slow
convergence. However we find it to be quite useful
if some form of regularization is used such
as spatially smoothing the gradient at each iteration.
Note the simplicity in implementing equation 2.9, and only requires
O(N2) algebraic operations
per iteration compared to O(N3) operations for the direct method.
Below is the MATLAB code for a steepest descent tomography algorithm.